Activities This Week

Let E be a quadratic imaginary field and p a prime which is inert in E. Let S be the special fiber (at p) of a unitary Shimura variety of signature (n,m) and hyperspecial level subgroup at p, associated with E/Q.

We study a natural foliation in the tangent bundle of S, which is originally defined on the \mu-ordinary stratum only, but is extended to a certain non-singular blow-up of S. We identify the quotient of S by the foliation with a certain irreducible component of a Shimura variety with parahoric level structure at p. As a result we get new results on the singularities of the latter.

We study integral submanifolds of the foliation and end the talk with a new conjecture of Andre-Oort type.

Hiroki Matui defined the concept of the topological group of an étale groupoid. In the case where the groupoid comes from a one-sided shift of finite type, he proved that the associated topological full group is finitely generated. We will show how to re-interpret these groups as tree almost automorphism groups and thus obtain compactly generated groups almost acting on trees.

This is the second survey talk in the series. See attached file.

In 1977 Sigmund proved that the space of ergodic measures supported on a basic set of an Axiom A diffeomorphism is path connected. In the talk I will describe a substantial generalization of this result to the space of hyperbolic ergodic measures supported on an isolated homoclinic class of a general diffeomorphism. Such homoclinic classes should be viewed as basic structural elements of any dynamics. Examples will be discussed. This is a joint work with A. Gorodetsky.